Rectangles within square

In order to make a rectangle, we need to pick two vertical and two horizontal lines that aren't the same distance apart.

There are $\frac{6*5}{2}=15$ ways to pick either two vertical or two horizontal lines, and these two lines range between 1 to 5 small square distances apart. For each range of one pair of vertical lines there's an equal amount of pairs of horizontal lines that we shouldn't count.

This makes the total number of rectangles: $5 \times (15-5)+4 \times (15-4)+3 \times (15-3)+2 \times (15-2)+1 \times (15-1)=170$ .

As a general rule,

Number of rectangles in an n*n grid is given by $Sigma$ $n^{3}$ ... (1)

&

Number of squares in an n*n grid is given by $Sigma$ $n^{2}$ .... (2)

Note that here n = 5

Since squares is a $subset$ of rectangles, therefore the number of rectangles is equal to the number of squares + number of rectangles that are not squares

On subtracting (2) from (1), we get the number of rectangles that are not squares

Total quadrilaterals are (1+2+3+4+5)^2 =225 . Square are 1^2+2^2+3^2+4^2+5^2=55.Therefore rectangles without square are 225-55=170

For 1 layer, 2 layers, 3 layers and 4 layers:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13

20  
15  
10  
5   

12  
8   
4   

6   
3   

2   

Horizontal + Vertical = (20 + 15 + 10 + 5 + 12 + 8 + 4 + 6 + 3 + 2) $\times$ 2 = 170.

Answer: $\boxed{170}$

Number of total rectangles - number of squares